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Nuclear Physics B

A new approach to the SU(2) ⊗ SU(2) symmetry of the Kepler motion

F. Ravndal, Takanari Toyoda

Published Oct 1, 1967• 25 citations
Publisher
DOI
10.1016/0550-3213(67)90081-8

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The coherent states for the quantum Kepler motion proposed in our previous work [Phys. Rev. A 59, 1021 (1999)], which is based on the dynamical $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ symmetry and the Duru-Kleinert auxiliary time, are improved by making use of the theory of the conserved-charge coherent states formulated by Bhaumik et al. and by Skagertsam. The expectation values for the angular momentum and the Runge-Lenz-Pauli vector with respect to the improved Kepler coherent states are also discussed.
Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the first case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, defined for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz, are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.
Fock (1935) explained the degeneracy of the energy levels of the Kepler problem (or hydrogen atom) in terms of the dynamical symmetry group SO(4). He showed that the problem is equivalent to the geodesic flow on the sphere S 3 . The 'hidden' symmetry SO(4) is made manifest. The classical n-dimensional Kepler problem can be better understood by enlarging the phase space of the geodesical motion on S n and including time and energy as canonical variables: a following symplectomorphism transforms the motion on S n in the Kepler problem. The authors prove that the Fock procedure is the implementation at 'quantum' level of the above-mentioned symplectomorphism.